CONTRACTS AS REFERENCE POINTS*
Oliver Hart and John Moore
July 2006, revised March 2007
We argue that a contract provides a reference point for a trading relationship: more
precisely, for parties’ feelings of entitlement. A party’s ex post performance depends on whether
he gets what he is entitled to relative to outcomes permitted by the contract. A party who is
shortchanged shades on performance. A flexible contract allows parties to adjust their outcome
to uncertainty, but causes inefficient shading. Our analysis provides a basis for long-term
contracts in the absence of noncontractible investments, and elucidates why “employment”
contracts, which fix wage in advance and allow the employer to choose the task, can be optimal.
* An early version of this paper was entitled “Partial Contracts.” We are particularly indebted to
Andrei Shleifer and Jeremy Stein for useful comments and for urging us to develop Section V.
We would also like to thank Philippe Aghion, Jennifer Arlen, Daniel Benjamin, Omri BenShahar, Richard Craswell, Stefano Della Vigna, Tore Ellingsen, Florian Englmaier, Edward
Glaeser, Elhanan Helpman, Ben Hermalin, Louis Kaplow, Emir Kamenica, Henrik Lando, Steve
Leider, Jon Levin, Bentley MacLeod, Ulrike Malmendier, Sendhil Mullainathan, Al Roth,
Jozsef Sakovics, Klaus Schmidt, Jonathan Thomas, Jean Tirole, Joel Watson, Birger Wernerfelt,
two editors and three referees for helpful suggestions. In addition we have received useful
feedback from audiences at the Max Planck Institute for Research on Collective Goods in Bonn,
the Harvard-MIT Organizational Economics Seminar, the University of Zurich, the 2006
Columbia University Conference on The Law and Economics of Contracts, Cornell University
(Center for the Study of Economy & Society), Yale University Law School (where part of the
text formed the first author’s Raben Lecture), Edinburgh University, London School of
Economics, the American Law and Economics Association Meetings (where part of the text
formed the first author’s Presidential address), Copenhagen Business School, the University of
Stockholm, Stockholm School of Economics, and the European Economic Association 2006
Annual Congress (where part of the text formed the second author’s Schumpeter Lecture). We
are grateful to Paul Niehaus for excellent research assistance. We acknowledge financial support
from the U.S. National Science Foundation through the National Bureau of Economic Research,
and the U.K. Economic and Social Research Council.
I. INTRODUCTION
What is a contract? Why do people write (long-term) contracts? The classical view held
by economists and lawyers is that a contract provides parties with a set of rights and obligations,
and that these rights and obligations are useful, among other things, to encourage long-term
investments.1 In this paper we present an alternative, and complementary, view. We argue that a
contract provides a reference point for the parties’ trading relationship: more precisely for their
feelings of entitlement. We develop a model in which a party’s ex post performance depends on
whether the party gets what he is entitled to relative to the outcomes permitted by the contract.
A party who is shortchanged shades on performance, which causes a deadweight loss. One way
the parties can reduce this deadweight loss is for them to write an ex ante contract that pins down
future outcomes very precisely, and that therefore leaves little room for disagreement and
aggrievement. The drawback of such a contract is that it does not allow the parties to adjust the
outcome to the state of the world. We study the trade-off between rigidity and flexibility. Our
analysis provides a basis for long-term contracts in the absence of noncontractible relationshipspecific investments, and throws light on why simple “employment” contracts can be optimal.
To motivate our work, it is useful to relate it to the literature on incomplete contracts. A
typical model in that literature goes as follows. A buyer and seller meet initially. Since the
future is hard to anticipate, they write an incomplete contract. As time passes and uncertainty is
resolved, the parties can and do renegotiate their contract, in a Coasian fashion, to generate an ex
post efficient outcome. However, as a consequence of this renegotiation, each party shares some
of the benefits of prior (noncontractible) relationship-specific investments with the other party.
Recognizing this, each party underinvests ex ante. The literature studies how the allocation of
asset ownership and formal control rights can reduce this underinvestment.2
1
While the above literature has generated some useful insights about firm boundaries, it
has some shortcomings.3 Three that seem particularly important to us are the following. First,
the emphasis on noncontractible ex ante investments seems overplayed: although such
investments are surely important, it is hard to believe that they are the sole drivers of
organizational form. Second, and related, the approach is ill-suited for studying the internal
organization of firms, a topic of great interest and importance. The reason is that the Coasian
renegotiation perspective suggests that the relevant parties will sit down together ex post and
bargain to an efficient outcome using side payments: given this, it is hard to see why authority,
hierarchy, delegation, or indeed anything apart from asset ownership matters. Finally, the
approach has some foundational weaknesses.4
We believe that in order to develop more general and compelling theories of contracts
and organizational form it is essential to depart from a world in which Coasian renegotiation
always leads to ex post efficiency.5 The purpose of our paper is to move in this direction. To
achieve this goal we depart from the existing literature in two key ways. First, we drop the
assumption made in almost all of the literature that ex post trade is perfectly contractible.
Instead we suppose that trade is only partially contractible.6 Specifically, we distinguish
between perfunctory performance and consummate performance, that is, performance within the
letter of the contract and performance within the spirit of the contract. 7 Perfunctory performance
can be judicially enforced, while consummate performance cannot.8 Second, we introduce some
important behavioral elements. We suppose that a party is happy to provide consummate
performance if he feels that he is getting what he is entitled to, but will withhold some part of
consummate performance if he is shortchanged—we refer to this as “shading.” An important
assumption we will make (for most of the paper) is that a party’s sense of entitlement is
2
determined by the contract he has written. This is the sense in which a contract is a “reference
point.” A companion assumption, also significant, is that the contract in question is negotiated
under relatively competitive conditions. A final element of the story is that there is no reason
why parties’ feelings of entitlement should be consistent. In particular, when the contract
permits more than one outcome, each party may feel entitled to a different outcome.
These ingredients yield the above-described trade-off between flexibility and rigidity. A
flexible contract has the advantage that parties can adjust the outcome to the state of the world,
but the disadvantage that any outcome selected will typically cause at least one party to feel
aggrieved and shortchanged, which leads to a loss of surplus from shading. An optimal contract
trades off these two effects. Our theory explains not only why parties will write somewhat rigid
contracts, but also the nature of the rigidity. The parties are more likely to put restrictions on
variables over which there is an extreme conflict of interest, such as price, than on variables over
which conflict is less extreme, such as the nature or characteristics of the good to be traded.
Among other things, our model shows why simple employment contracts, which fix price (wage)
in advance and allow the employer to choose the task, can be optimal. (More generally, the
model explains why the wage should vary with the task if some tasks are systematically costlier
than others.)
For most of the paper we suppose that parties’ feelings of entitlement are controlled
entirely by the contract they have written. In reality other influences on entitlements are
sometimes important. For example, parties may look to related transactions to determine
whether they are being fairly treated. This consideration allows for a rich new set of
possibilities; we examine these briefly in Section V. Although our analysis is preliminary, we
show that external measures of entitlement can interfere with an ex ante contract, and that it may
3
therefore be optimal for the parties to postpone contracting, i.e., the optimal ex ante contract may
be “no contract.”
The behavioral assumptions on which our analysis is based are undoubtedly strong, and
although they are broadly consistent with a number of ideas in the literature, there is no single
model or experiment that we can appeal to that supports precisely what we do. In future work it
would therefore be highly desirable to see whether our assumptions can be validated by
experiment. At the same time we should make it clear that we are not wedded to a particular set
of behavioral assumptions. In addition, and relatedly, we see the behavioral approach adopted
here as something of a means to an end; the end is the development of a tractable model of
contracts and organizational form that exhibits ex post inefficiency, and that can explain simple
contracts observed in reality, such as the employment contract. From this perspective the costs
of flexibility that we focus on – shading costs – can be viewed as a short-hand for other kinds of
transaction costs, such as rent-seeking, influence, and haggling costs. We return to this theme in
the conclusions.
The paper is organized as follows. Section II presents the model, discusses our key
assumptions, and lays out a simple example. In Section III we analyze a case where there is
uncertainty about value and cost but not about the type of good to be traded. In Section IV we
consider a second case where there is uncertainty about the nature of the good. This section also
discusses the employment relationship. Section V allows for the possibility of influences on
entitlements other than the initial contract. In Section VI we discuss renegotiation. Finally,
Section VII concludes. The Appendix considers a more general class of contracts than those
studied in the text and includes proofs of theorems.
4
II. THE MODEL
We consider a buyer B and a seller S who are engaged in a long-term relationship. The
parties meet at date 0 and can trade at date 1. We assume a perfectly competitive market for
buyers and sellers at date 0, but that competition is much reduced at date 1: in fact, for the most
part we suppose that B and S face bilateral monopoly at date 1. In other words, there is a
“fundamental transformation” in the sense of Williamson [1985].
We do not model why this fundamental transformation occurs. It could be because the
parties make relationship-specific investments, but there may be other more prosaic reasons. For
example, imagine that B is organizing a wedding for his daughter. S might be a caterer. Six
months before the wedding, say, there may be many caterers that B can approach and many
weddings that S can cater. But it may be very hard for B or S to find alternative partners a week
before the wedding. While there are no very obvious relationship-specific investments here, the
fundamental transformation seems realistic, and the model applies.
It would be easy to fit relationship-specific investments explicitly into the analysis, but
we would then suppose that these investments are contractible. That is, an important feature of
our model is that it does not rely on noncontractible investments.
We make some standard assumptions. Any uncertainty at date 0 is resolved at date 1.
There is symmetric information throughout, and the parties are risk neutral and face no wealth
constraints.
We now come to the two assumptions that represent significant departures from the
literature. First we suppose that ex post trade is only partially contractible. Specifically, while
the broad outlines of ex post trade are contractible, the finer points are not. As noted in the
Introduction, we distinguish between perfunctory and consummate performance, that is,
5
performance within the letter of the contract and performance within the spirit of the contract.
Perfunctory performance is enforceable by a court while consummate performance can never be
judicially enforced.
For instance, in the wedding example, a judge can determine whether food was provided,
but not the quality of the cake or whether the host was friendly to the catering staff.
Before we describe our second (set of) assumption(s), it is useful to provide a time line;
see Figure I. The parties meet and contract at date 0. At this stage there may be uncertainty and
so the parties typically choose to write a flexible contract that admits several outcomes. At date
1 the uncertainty is resolved and the parties refine the contract, that is, they decide which
outcome to pick. After this, trade occurs and the degree of consummate performance is
determined.9
[FIGURE I HERE.]
We now come to our second key departure from the literature. We make a number of
assumptions – some behavioral – about the determinants of consummate performance. First, we
suppose that consummate performance does not cost significantly more than perfunctory
performance: either it costs slightly more or it costs slightly less, that is, a party may actually
enjoy providing consummate performance. In what follows, to simplify matters, we assume that
a party is completely indifferent between providing consummate and perfunctory performance.
Given this indifference we take the view that a party will be willing to provide
consummate performance if he is “well treated,” but not if he is “badly treated.”10 We make the
crucial assumption that a party is “well treated” if and only if he receives what he is entitled to;
and that the date 0 contract acts as a reference point for entitlements. In fact, for most of the
paper we suppose that the contract is the sole reference point for entitlements (but see Section
6
V). What we mean by this is that a party does not feel entitled to more than the best outcome
permitted by the contract. So, for example, if the date 0 contract specifies just one outcome, then
each party will feel that he is getting exactly what he is entitled to if that outcome occurs.11 We
discuss the assumption that the contract acts as a reference point further below. As we shall
make clear, this assumption is linked to a companion assumption that the contract is negotiated
under relatively competitive conditions.12
Matters become more complicated if the contract specifies more than one outcome. Now
we take the view that the parties may no longer agree about what they are entitled to. In
particular, if the contract says that either outcome a or outcome b can occur, then one party may
feel entitled to a and the other to b. We do not model why these differences in entitlements arise,
but we have in mind the kinds of effects described in the self-serving bias literature.13 To
capture conflict as simply as possible we suppose that each party feels entitled to the best
outcome permitted by the contract. However, as Section VII will make clear, our analysis does
not depend on such an extreme view of entitlements.
The final piece of the story is that getting less than what you are entitled to causes
aggrievement and leads to retaliation and “shading,” i.e., stinting on consummate performance.
Let uB, uS denote the buyer and seller’s gross payoffs from the actual contractual outcome.
Define the buyer’s aggrievement level aB to equal the maximum gross payoff he could have
achieved, taken over all contractually feasible outcomes, minus uB. The seller’s aggrievement
level aS is defined analogously. (See Section III and the Appendix for more detail.) Denote by
σB the monetary loss that the buyer imposes on the seller through shading; analogously, σS is the
loss the seller imposes on the buyer. Then we suppose that the buyer and seller’s net payoffs can
be written as
7
(1)
UB = uB – σS – Max{θ aB – σB, 0},
(2)
US = uS – σB – Max{θ aS – σS, 0},
where 0 < θ ≤ 1. The last term in (1) and (2) captures the idea that aggrievement of $1 causes a
direct psychic loss to the party experiencing it of $θ, but that the party can offset this by shading,
i.e., in effect by transferring the hurt back to the other party, up to the point where the
aggrievement disappears. Given (1) – (2), the buyer will choose σB = θ aB and the seller σS =
θ aS, and so
(3)
UB = u B – θ a S ,
US = uS – θ aB.
Note that implicit in our formulation is the assumption that the buyer and seller
experience aggrievement equally, have the same ability to transfer it to the other party, and that
there is no upper bound on shading; in effect θ represents both the desire and ability to shade.
We discuss this further below.14
The shading decisions σB, σS are made simultaneously by B and S, and are not observable
to an outsider. Hence they are not contractible, even at date 1 (recall our assumption that
consummate performance can never be enforced judicially). We also suppose that shading is
infeasible if the parties do not trade at date 1 (trade can be shaded but no trade cannot be
shaded).15
8
It may be useful to give some examples of shading. There are many ways one trading
partner can hurt another. For example, a seller can shade by cutting quality, e.g., in the wedding
example, she can stint on some of the ingredients of the wedding cake. Or the seller may
withhold cooperation. The buyer may want the seller to turn up half an hour early. The cost to
the seller may be low, and she would normally oblige. But if the seller feels aggrieved she may
refuse this request at considerable cost to the buyer. A third example would be “working to
rule”: the seller abides by the strict terms of the contract and offers no more.16
Buyers can also shade. While it is harder to imagine a buyer cutting back on quality, it is
easy to think of situations where a buyer refuses to make minor concessions or to cooperate (for
example, the buyer may turn down the seller’s request to come half an hour later). The buyer
can also make life difficult for the seller by quibbling about details of performance or by
delaying payment or by giving a bad reference. Thus, while the assumption that B and S can
shade symmetrically (e.g., have the same parameter θ) is strong, we view it as a natural starting
point for our analysis. As we will discuss in Section VII, we do not believe that this assumption
is crucial. For example, a model in which the buyer experiences aggrievement, as in equation
(1), but is forced to “eat” the aggrievement rather than being able to shift it to the seller, i.e., σB ≡
0, is likely to yield similar results.
At this point, it is useful to illustrate the model with a simple example. Suppose that B
requires one unit of a standard good – a widget – from S at date 1. Assume that it is known at
date 0 that B’s value is 100 and S’s cost is zero: there is no uncertainty. What is the optimal
contract?
The answer found in the standard literature is that, in this setting without noncontractible
investments, no ex ante contract is necessary: the parties can wait until date 1 to contract. To
9
review the argument, imagine that the parties do wait until date 1. Assume that Nash bargaining
occurs and they divide the surplus 50 : 50, i.e., the price p = 50. Of course, a 50 : 50 division
may not represent the competitive conditions at date 0. For simplicity, suppose that there is one
buyer and many sellers at date 0, so that in competitive equilibrium B receives all the surplus.
Then S will make a lump-sum payment of 50 to B at date 0: in effect S pays B up front for the
privilege of being able to hold B up once the parties are in a situation of bilateral monopoly.
This “no contract” solution, combined with a lump-sum payment, no longer works in our
context. To see why, suppose for the moment that “no contract” means that trade can occur at
any price between zero and 100 (we revisit this assumption in Section V). (Prices above 100 are
irrelevant since B will reject the widget and prices below zero are irrelevant since S will refuse to
supply.) But this means that when the parties reach date 1 there is much to argue about.17 The
best contractual outcome possible for B is a zero price and our assumption is that he will feel
entitled to it; and the best contractual outcome possible for S is a price of 100 and our
assumption is that she will feel entitled to it. In spite of these conflicting feelings of entitlement,
the parties will settle on some price p between 0 and 100, and trade will occur. However, each
party will feel aggrieved and will shade. Since B feels aggrieved or shortchanged by p, he
shades by θp, and since S feels aggrieved or shortchanged by (100 – p), she shades by θ(100 –
p). Thus the final payoffs are
(4)
UB = 100 – p – θ(100 – p) = (1 – θ)(100 – p),
(5)
US = p – θp = (1 – θ)p,
10
and total surplus is given by
(6)
W = (1 – θ)100.
We see that, independent of p, there is a loss of 100 θ.18
What can be done to eliminate this loss? The first point to note is that ex post Coasian
renegotiation at date 1 does not do the job. The reason is that shading is not contractible, and
thus a contract not to shade is not enforceable. To put it another way, if B offers to pay S more
not to shade, then while this will indeed reduce S’s shading since S will feel less aggrieved, it
will increase B’s shading because B will feel more aggrieved! In fact, it is clear from (6) that
changes in p do not affect aggregate shading, which is given by 100 θ.
Note that the conclusion that the loss from shading equals 100 θ remains true even if the
parties replace renegotiation at date 1 by a mechanism. For example, suppose B and S agree at
date 0 that B will make a take-it-or-leave-it offer to S at date 1. The best offer for B to make is p
= 0. However, S will feel that B could and should have offered p = 100 since S is entitled to this.
Thus S will be aggrieved by 100, and will shade by 100 θ. Hence the loss from shading is again
100 θ.
Although these approaches do not work, there is a very simple solution to the shading
problem. The parties can write a contract at date 0 that fixes p at some level between 0 and 100,
e.g., if there are many sellers and only one buyer at date 0, then it would be natural to set p = 0.
Then there is nothing to argue about at date 1. Neither party will feel aggrieved or shortchanged
since each receives exactly what he or she bargained for and expected. Hence no shading occurs
and total surplus equals 100.19
11
As we have argued earlier, a contract that fixes price works because it anchors the
parties’ expectations and feelings of entitlement: the contract is a reference point. An obvious
question to ask is, what changes between dates 0 and 1? Why does a date 0 contract that fixes p
avoid aggrievement, whereas a date 1 contract that fixes p does not? Our view is that the ex ante
market plays a crucial role here. Since the date 0 market is more competitive than the date 1
market it provides a relatively objective measure of what B and S bring to the relationship. For
example, if the date 0 market is perfectly competitive and the market price is p̂ , then S accepts
that p̂ is a reasonable price because no seller receives more; and B accepts that p̂ is a reasonable
price because no seller is willing to supply for less. More generally, if the date 0 market is not
perfectly competitive – imagine that there is a range of prices [p, p ] in the “market,” where 0 < p
< p < 100 – then while B will feel entitled to p and S to p , the shading costs of aggrievement
will be θ ( p - p); this is less than the shading costs of 100 θ that would arise if the contract is left
until date 1.
Note that we are supposing that the existence of the date 0 market is not enough: B and S
must embody the information from this market in a date 0 contract. We take the view that, if B
and S pass up the opportunity to write a contract at date 0, then by the time date 1 arrives, the
date 0 market has no particular salience, self-serving biases about individual values can come
into play [see footnote 13], and the result will be argument, aggrievement, and shading. In
effect, we are supposing that the contractual process itself is a key force in anchoring
entitlements.
To the extent that the role of the contract is to embody and anchor entitlements, the fact
that the contract is legally binding is perhaps of secondary importance. Much of our analysis
goes through if the contract is viewed as a non-binding agreement.20 See in particular the
12
discussion of agreements to agree in Section III. Note, however, that the solemnity that
accompanies the writing of a legally binding contract may help to give weight to the expectations
and entitlements embodied in that contract.
The example analyzed in this section is very special because a date 0 contract that fixes
price achieves the first-best. The first-best is no longer achievable if either (a) v, c are uncertain;
or (b) the nature of the good (the widget) is uncertain. We study case (a) in Section III and case
(b) in Section IV.
III. THE CASE WHERE VALUE AND COST ARE UNCERTAIN
In this section we consider the case where B wants one unit of a standard good – a widget
– from S at date 1 but there is uncertainty about B’s value v and S’s cost c. This uncertainty is
resolved at date 1. There is symmetric information throughout, so that v, c are observable to
both parties. However, v, c are not verifiable, and so state-contingent contracts cannot be
written.
We make an important simplifying assumption. We suppose that trade occurs at date 1 if
and only if both parties want it, i.e., trade is voluntary. To put it another way, if no trade occurs
an outsider (e.g., a judge) cannot tell whether this is because the seller refused to supply the
widget or the buyer refused to accept it.21 As a result, a party cannot be punished for breach of
contract. We are confident that the main ideas of this section generalize to the case where
specific performance is possible, but the details become more complicated.
Note that the model can also be interpreted as applying to the case where the parties write
an “agreement to agree” at date 0. That is, suppose that the parties intend to use the date 0
agreement as a framework for future negotiation (this corresponds to the refinement process in
13
Figure I), but for some reason are not yet ready to sign a binding contract. The usual legal
presumption is that either party can opt out of such an agreement if future negotiations fail. Thus
the voluntary trade assumption holds. 22
In this setting the simplest kind of contract consists of a no-trade price p0 and a trade price
p1. Given the voluntary trade assumption, trade will occur (q = 1) if and only if
(7)
v ≥ p1 – p0 ≥ c.
From (7) it is clear that only the difference between p1 and p0 matters, and so, given the existence
of lump-sum transfers, we can normalize p0 to be zero.23
It is worth comparing (7) to the first-best trading rule, given by
(8)
q = 1 Ù v ≥ c.
We need to deal with one further issue before we proceed. After the uncertainty about v,
c is resolved, suppose v > c but either v < p1 – p0 or c > p1 – p0. At this stage, the parties might
want to renegotiate their contract. Renegotiation does not fundamentally change our results and
so, for the moment, we ignore it; we return to it in Section VI.
Since we want to allow for contractual flexibility we shall wish to generalize beyond
simple contracts. One way to introduce flexibility is to suppose that the contract specifies a notrade price p0 and an interval of trading prices [p, p ]. Suppose for simplicity that B chooses the
trade price at date 1. Then
14
(9)
q = 1 Ù ∃ p ≤ p1 ≤ p s.t. v ≥ p1 – p0 ≥ c.
In other words, trade occurs if and only if B can find a price in the range [p, p ] so that the
parties want to trade (B will choose the lowest such price). Actually, it is clear that the same
trading rule (9) holds if S chooses p1 (S will choose the highest price in the range [p, p ] that
guarantees trade); moreover, the level of shading will be the same given that the parties have the
same θ. This feature – that the mechanism for choosing the outcome doesn’t matter – is special
to the model of this section: it will importantly not hold in the model of Section IV.
It follows from (9) that, again, only the difference between p1 and p0 matters, and so we
can normalize p0 ≡ 0 and rewrite (9) as
(10)
q = 1 Ù v ≥ c, v ≥ p, c ≤ p .
More general contracts than p0 ≡ 0, p ∈ [p, p ] are in fact possible. For example, a
contract could permit p to lie in some set other than an interval, or it could allow p0 and p1 both to
vary. In the Appendix we show that (with a slight refinement of our assumptions) our main ideas
extend to the case where a contract consists of an arbitrary set of (p0, p1) pairs and a mechanism
– a game – for choosing among them.
Given a contract [p, p ] (that is, p0 ≡ 0, p ∈ [p, p ]), what determines aggrievement? Our
hypothesis is that each party feels entitled to the best outcome permitted by the contract.
However, each party also recognizes that, given the voluntary trade assumption, he or she cannot
hope to obtain more than one hundred percent of the gains from trade. This means that S feels
15
entitled to p = Min (v, p ) and B feels entitled to p = Max (c, p). (Another way to think about it
is that if S had complete control over the price but had to stick within the contract she would
choose p = Min (v, p ), and if B had complete control over the price but had to stick within the
contract he would choose p = Max (c, p).) Thus aggregate aggrievement equals {Min (v, p ) –
Max (c, p)}. An optimal contract maximizes expected surplus net of shading costs. (Lump-sum
transfers are used to reallocate surplus.) Thus an optimal contract solves:
(11)
Max
p,p
∫ [v - c - θ {Min(v, p ) – Max(c,p)}]dF(v,c),
v ≥c
v≥ p
c≤ p
where F is the distribution function of (v, c).
The trade-off is clear. A large interval [p, p ] makes it more likely that trade will occur if
v ≥ c. (If p = -∞, p = ∞, the trading rule becomes the first-best one: q = 1 Ù v ≥ c.) However, it
also increases expected shading costs.
We refer to a contract where p = p as a simple contract, and a contract where p < p as a
non-simple contract. We start off with some cases where the first-best is achievable with a
simple contract.
PROPOSITION 1. A simple contract achieves the first-best if (i) only v varies; (ii) only c
16
varies; (iii) the smallest element of the support of v is at least as great as the largest
element of the support of c.
The proof of Proposition 1 is immediate. If only v varies, choose a simple contract with p
= c. If only c varies, choose a simple contract with p = v. If (iii) holds, choose a simple contract
with p between the smallest v and largest c.
In some cases one needs a non-simple contract to achieve the first-best.
EXAMPLE 1.
Suppose that there are two states of the world. In s1, v =9, c = 0. In s2, v = 20,
c = 10. In other words, either v and c are both low or they are both high.
s1
s2
v
9
20
c
0
10
Obviously, one cannot get the first-best with a simple contract since there is no
price p that lies both between 0 and 9 and between 10 and 20. However, a contract that
specifies an interval of trading prices [9,10] (p = 9, p = 10), with B choosing the price,
does achieve the first-best. To see why, note that in s1 B will choose p = 9 since this is
the lowest available price. S will not be aggrieved since, even if S could choose the
price, she would not pick a price above 9 given that this would cause B not to trade. In s2
B picks p = 10 since this is the lowest price consistent with S being willing to trade. S is
17
again not aggrieved since she couldn’t hope for a higher price than 10 given that 10 is the
highest available price. Thus, the contract p = 9, p = 10 achieves trade in both states
without any shading.
Note that in this example the optimal contract is unique. Any price range smaller than
[9,10] would fail to generate trade in one of the states, and any price range larger than [9,10]
would cause aggrievement in at least one of the states.
We now turn to an example where the first-best cannot be achieved even with a nonsimple contract.
EXAMPLE 2
The example is the same as the previous one except that there is a third state, s3,
where v is high and c is low.
s1
s2
s3
v
9
20
20
c
0
10
0
The first-best cannot be achieved because, in order to ensure trade in s1, s2, we
need p ≤ 9, p ≥ 10. But such a price range leads to aggrievement and shading in s3.
There are three possible candidates for a second-best optimal contract:
(a) p = p = 9.
18
This contract yields trade in s1 and s3 but not in s2. Since there is nothing to argue about
– the price is fixed at 9 – there is no shading. Total surplus is given by
Wa = 9 π1 + 20 π3 ,
where π1, π3 are the probabilities of s1, s3, respectively.
(b) p = p = 10.
This contract yields trade in s2 and s3 but not in s1. Since there is nothing to argue about
– the price is fixed at 10 – there is no shading. Total surplus is given by
Wb = 10 π2 + 20 π3,
where π2 is the probability of s2.
(c) p = 9, p = 10.
This contract yields trade in all three states, but there is aggregate aggrievement of 1 in
s3. Total surplus is given by
Wc = 9 π1 + 10 π2 + (20 – θ) π3.
19
Obviously, which of these contracts is optimal depends on the probabilities π1, π2,
π3 and θ. Contract (a) is optimal if π2 is small, contract (b) is optimal if π1 is small, and
contract (c) is optimal if π3 or θ is small.
The reader may wonder whether more general contracts than p0 ≡ 0, p ∈ [p, p ] can do better. In
particular, can “Maskin mechanisms” help?24 Maskin mechanisms are a way, in effect, of
making observable information verifiable. Note that if the state were verifiable, it would be easy
to achieve the first-best. For example, a contract that specifies p = 9 in s1 and s3, and p = 10 in
s2 would do the job. Call this contract (d).
In the Appendix we show that more general contracts do not help in our situation.
Suppose that, as in a Maskin mechanism, each party reports the state of the world. If they agree
the price is as in contract (d), say. If they disagree something unpleasant happens. The problem
is that in s3 S would like B to play the mechanism as if it were s2 and will be aggrieved by 1 if B
refuses to go along with this. On the other hand B will be aggrieved by 1 if S refuses to play the
mechanism as if it were s3. Either way aggregate aggrievement in s3 is 1, which yields total
surplus equal to Wc, as in contract (c). The general point is that under our assumptions
aggrievement is determined by the entire set of terminal nodes of a mechanism, not just the
equilibrium ones (see the Appendix for more details).25
It may be useful to talk more generally about state contingency (or, more accurately, lack
of state contingency). Why can’t agreements between two parties who will both learn the state
of the world be made state contingent? In principle, one could imagine B and S having the
following conversation at date 0 about contract (c). B could tell S that he will pay 9 at date 1 in
20
all circumstances unless S will not trade at this price; in which case B will raise the price to 10.
B could explain to S that she should not feel aggrieved in s3 when B does not raise the price to
10 since B said that he would stick with 9. To put it a little more formally, a contract that makes
price a function of an observable but unverifiable state of the world should not cause
aggrievement since both parties observe the state and can see whether the other party is sticking
to the contract.
There is no doubt that this argument has some force and indeed it is not uncommon for
contracting parties to have informal understandings about what to do if certain observable events
occur, e.g., it rains at the wedding. However, events like the weather are pretty objective –
indeed close to verifiable. We believe that state-contingent agreements are more problematic if,
as in Example 2, the state in question is a (more) subjective value-cost pair. Subjectivity opens
the door to differing interpretations. For example, between dates 0 and 1, S may convince
herself that her skills or (unmodeled) actions contribute greatly to the trading opportunity at date
1. S may feel that she deserves to be rewarded for this (self-serving bias may be at work).
Whatever speech B has made at date 0, B has the option, if he so chooses, to recognize S’s
contribution by raising the price to 10, i.e., this is consistent with the contract. In other words, B
can pretend that S’s cost is 10 even if it isn’t. If B refuses to recognize S’s contribution and
offers only 9, S will take this to be an ungenerous act and will respond by shading.26
This brings to a close our discussion of the case where parties induce flexibility by
specifying a price range.27 It is not clear how common this case is. One reason is that in practice
the parties may be able to ensure trade when v and c vary through specific performance. Note,
however, that price ranges are observed in the case of agreements to agree.28 In any event, we
see the model of this section as something of a five-finger exercise. In the next section we
21
consider a model where the uncertainty concerns the nature of the good to be provided. We will
show that this model can shed light on the employment relationship.
IV. THE CASE WHERE THE NATURE OF THE GOOD IS UNCERTAIN
In this section we consider the case where there is uncertainty about the nature of the
good or service B requires from S. For example, S might provide secretarial services for B, and
B may not know in advance whether he wants S to type letters or file papers. We will actually
use a more colorful example. We will suppose that B is arranging an evening with friends and
wants S to perform music. The nature of the music may depend on eventualities that will occur
between dates 0 and 1, e.g., who is coming to the evening, what music S is rehearsing for other
performances, etc.
To make matters as simple as possible, we will assume that there are two types of
music/composers that it might be efficient for S to play: Bach and Shostakovich. In the
Appendix we also allow for convex combinations of Bach and Shostakovich, but in the text we
will not need to do this. Each composer can take on one of two value-cost combinations, given
by (v, c) and (v – Δ, c – δ), respectively, where v > v – Δ > c > c – δ, i.e., the value-cost supports
of the composers overlap. (Everything is measured in money terms.) In other words a composer
can be “high value-high cost” or “low value-low cost.” We do not insist on stochastic
independence of the two value-cost combinations, but we do impose symmetry, i.e., the
probability that Bach is “high value-high cost” and Shostakovich is “low value-low cost” is the
same as the probability of the reverse. Thus, there are four states of the world:
[FIGURE II HERE.]
22
We start with the case Δ > δ. This implies that the high value-high cost composer yields
more surplus than the low value-low cost composer and should be chosen whenever available.
Thus the first-best has any music in states s1 and s4, Bach in s2, and Shostakovich in s3.
Expected total surplus is W = v – c – π4 (Δ – δ).
What is the optimal second-best contract given that the state is observable but not
verifiable? We continue to assume voluntary trade and set p0 ≡ 0. We also focus on contracts
that deliver symmetric outcomes, i.e., whatever composer occurs in s2, the “mirror image”
composer occurs in s3, and the prices are the same in the two states.
It will simplify the presentation to start with the case π1 = π4 = 0, i.e., where s2 and s3
each occur with probability ½. We will see that this case is actually too simple, but it is useful
for building up intuition.
There are four natural candidates for an optimal contract: no contract, a contract that
fixes price and lets B choose the composer, a contract that fixes price and lets S choose the
composer, and a contract that fixes price and composer. We consider these in turn.
(a) No contract
The parties can always wait until date 1 to contract, by which time they will know whether s2 or
s3 has occurred. They will then bargain over the division of surplus, v – c. The analysis is
similar to that in Section II: whatever price between c and v is agreed to, the total amount of
aggrievement will be (v – c), and so shading costs will be θ(v – c). Hence net surplus is
Wa = (1 – θ) (v – c).
23
(b) A contract that fixes the price p such that c ≤ p ≤ v – Δ and lets B choose the composer at
date 1
Given that price is fixed, B will choose the highest value composer at date 1: Bach in s2,
Shostakovich in s3. Since Δ > δ this is the efficient choice. However, in each state the seller
will be aggrieved that B did not choose her favorite composer: Shostakovich in s2, Bach in s3.
The seller will be shortchanged by δ, the difference in the costs between the two composers, and
so shading costs are θδ. Total surplus is
Wb = v – c – θδ.
(c) A contract that fixes the price p such that c ≤ p ≤ v – Δ and lets S choose the composer at
date 1
Given that price is fixed, S will be inclined to choose the lowest cost composer at date 1, which
is inefficient. B will be aggrieved by Δ, the difference in value between his favorite composer
and S’s choice, and will shade so that S’s payoff falls by θΔ. Note that if θΔ > δ, S will be worse
off than if she had chosen the high cost composer, and so she will choose the high cost composer
after all; in effect, we are back to contract (b) (with S being aggrieved by δ because she really
wants to choose the low cost composer). On the other hand, if θ Δ < δ, S will stick with the low
cost choice, and surplus is
Wc = v – Δ – c + δ – θΔ.
24
(d) A contract that fixes the price p such that c ≤ p ≤ v – Δ and fixes the composer
Suppose the composer is fixed at Bach, say. This yields the efficient choice of composer half the
time, and so surplus is
Wd = ½(v – c) + ½(v – Δ – c + δ) = v – c – ½ Δ + ½ δ.
Note that there is no cost of shading in contract (d) since, with price and composer fixed, there is
nothing to be aggrieved about.
How do contracts (a) – (d) compare? Since v – Δ > c and Δ > δ, it is easy to see that Wb
> Wa. (If δ ~
− 0, Wb ~
− v – c, i.e., contract (b) achieves approximately the first-best.) Also it is
clear that Wb > Wc (in contract (c) the composer is less efficient than in (b) and there is more
shading). So the choice is between contracts (b) and (d). Algebra tells us that (b) is better if and
only if
(12)
Δ > (1 + 2θ) δ.
The analysis so far is a little misleading. There is a contract that performs better than
either (b) or (d), and in fact achieves the first-best. This contract is a variation of (b), in which
the constraint c ≤ p ≤ v – Δ is dropped. Specifically, consider the contract in which the price p =
v and B chooses the composer. B will make the efficient choice in each state (as in contract (b)),
but in addition S will not be aggrieved. The reason is that B exactly breaks even: any composer
25
who is more favorable to S than Bach in s2 and Shostakovich in s3 would cause B to refuse to
trade, i.e., such a composer would violate B’s individual rationality constraint.
There is something very fragile about the contract just described. If there is any chance
that B’s value falls below v it will lead to no trade. In the next proposition we return to the case
where states s1 and s4 have positive probability. The proposition tells us that, as long as π4 is not
too low, fixing p close to v is a bad idea. The proposition also requires π1 not to be too low in
order to rule out subtle contracts that can also do better than (b) and (d). In the Appendix we
establish formally:
PROPOSITION 2. Assume v > v – Δ > c > c – δ, and Δ > δ. Suppose in addition
π4
≥
1 − π1
π1
m
δ
, where m = min {θδ, ½(Δ – δ)}, and
. Then the
≥
(v - c - Δ + δ + m)
1−π4 Δ + δ
optimal second-best contract fixes c ≤ p ≤ v – Δ. In addition, if Δ > (1 + 2θ)δ, B is given
the right to choose the composer, while, if Δ < (1 + 2θ)δ, the parties fix the composer, at
Bach say.
Proposition 2 illuminates the different roles played by price and music in the model of
this section. Price has no allocative role – its choice is a zero sum game – and so, in order to
avoid aggrievement, it is better to fix it in advance. Music does serve an allocative role and so, if
Δ/δ is large or θ is small, it makes sense to leave it open. Moreover, if Δ/δ is large or θ is small,
B should choose the composer since B will make an efficient choice, and, given that S cares
relatively little, aggrievement will be low.
26
Note that there are two implicit assumptions underlying Proposition 2. First, aggregate
uncertainty is small (v – Δ > c), and so price does not have to vary across the four states in order
to ensure that both parties wish to trade, as it did in Section III. Second, there is no systematic
relationship between composer and cost. In contrast, if Shostakovich, say, was on average
costlier for S to play than Bach, then it would be optimal to have a higher price for Shostakovich
than Bach, in order to reduce S’s aggrievement in states where Shostakovich is chosen. In other
words, a generalized version of the model can explain why price should vary with the service S
provides if there are systematic differences in costs.29
Let’s now turn to the case where Δ < δ, i.e., the low value-low cost composer yields more
surplus than the high value-high cost composer. The argument goes through as above except that
now it is never optimal for B to choose the composer but it may be optimal for S to choose the
composer: Contract (c) may be optimal, but contract (b) is not. We have
PROPOSITION 3. Assume v > v – Δ > c > c – δ, and Δ < δ. Suppose in addition
π1
π4
m′
Δ
≥
, where m′ = min {θΔ, ½(δ – Δ)}, and
. Then the
≥
1 − π 4 (v - c + m ′)
1 − π1 δ + Δ
optimal second-best contract fixes c ≤ p ≤ v – Δ. In addition, if δ > (1 + 2θ) Δ, S is given
the right to choose the composer, while, if δ < (1 + 2θ) Δ, the parties fix the composer, at
Bach say.
We believe that Propositions 2 and 3 can throw light on a classic question: the nature of
the employment relationship and the difference between an employee and an independent
contractor. In early work, Coase [1937] and Simon [1951] argued that a key feature of the
employment relationship is that an employer tells an employee what to do. This view was
27
challenged by Alchian and Demsetz [1972], and the more recent literature has emphasized asset
ownership as the distinguishing aspect of these relationships [see Grossman and Hart, 1986, and
Hart and Moore, 1990].30 The current model allows us to return to the ideas of Coase and
Simon. We interpret the case where B chooses the composer as an employment relationship and
the case where S chooses the composer as independent contracting.31 That is, if B hires S’s
musical services for the evening, with the understanding that B will tell S what to do, then S is
working for B. In contrast, if B engages S to provide an evening of music, with the details of
exactly how this is to be done left up to S, then S is an independent contractor.32
Propositions 2 and 3 tell us that when θ is small, if B cares more about the composer, that
is, Δ > δ, then employment is better (if δ ~
− 0, an employment contract achieves approximately the
first-best); while if S cares more about the composer, that is, Δ < δ, then independent contracting
is better (if Δ ~
− 0, independent contracting achieves approximately the first-best). In both cases
it is optimal for the parties to fix the price in advance.
While these results are in the spirit of Coase and Simon, they differ from Simon’s formal
argument in important ways. Simon would also argue that B should choose the composer if B
cares more about the composer than S. However, in Simon’s model it is not clear why an ex ante
contract is needed at all. Since his model has neither aggrievement nor noncontractible
investment, the parties can rely on Coasian bargaining at date 1. Also, a contract that achieves
the first-best in Simon’s model is one where B has the right to make a take-it-or-leave-it offer to
S; i.e., B proposes a price-composer pair, and S can accept or reject it. In other words, in
Simon’s model there are many optimal contracts (a continuum, in fact), of which the
employment contract is just one.
28
We saw in Section II that this is not true in our model. For example, consider the
contract in which B offers a price-composer pair. B will suggest price equal to cost, and there
will be aggrievement and shading in all states since S will feel entitled to price equal to value.
Thus this contract performs strictly worse than the employment contract.
In other words a virtue of our model is that it can explain why, given θ > 0, the
employment contract is uniquely optimal when Δ > δ and θ is small; why independent
contracting is uniquely optimal when Δ < δ and θ is small; and why in all the cases considered in
this section it makes sense (in the absence of systematic cost differences across composers or
tasks) for the parties to fix price ex ante, i.e., to take price off the table.
V. EXTERNAL REFERENCE POINTS
So far we have assumed that a prior contract is the only reference point for the transaction
at date 1. In this section we relax this assumption. Our analysis is preliminary and speculative.
It is not difficult to think of situations where parties look outside a contract to determine
whether they are being treated fairly. A familiar case is where someone is hired as an employee
at a particular wage, and some time later someone else with comparable or even inferior skills is
hired by the same employer at a higher wage, perhaps because market conditions have changed.
The first person will almost certainly feel unhappy about this even though their wage was
determined fairly and competitively at the time.33
One way to capture the idea of “external” reference points is as follows. Return to the
model of Section III where the parties trade a standard good but there is uncertainty about v and
c. Suppose that in each state of the world there is a range of “reasonable” prices for the good,
determined exogenously, and given by [p min, p max]. The interpretation is that this range is
29
based on comparable transactions: trades in other markets at date 1, prices of previous
transactions, prices embodied in new contracts written between dates 0 and 1, etc. Any price
between p min and p max can be justified to outsiders as being reasonable while other prices
cannot.34
In order to simplify matters, we will assume that the [c, v] and [p min, p max] intervals
always intersect; that is, whenever v ≥ c, v ≥ p min and c ≤ p max. This assumption captures the
idea that external reference points and internal value and cost are never too far apart.
The [p min, p max] range plays two roles. First, the range may affect entitlements in the
presence of a contract. Second, the range may affect bargaining in the absence of a contract.
Consider the first role. Suppose that the parties’ date 0 contract specifies the range of trading
prices [p, p ]. We saw in Section III that, on the basis of this, S feels entitled to receive Min (v,
p ) and B to pay Max (c, p). We assume that the external reference points [p min, p max] modify
these entitlements only if (i) p min > Min (v, p ) or (ii) Max (c, p) > p max. In the first case the
price S feels entitled to receive is raised to p min; while in the second case the price B feels
entitled to pay is lowered to p max.
In other words, S feels entitled to receive more than Min (v, p ) if (and only if) all
external prices lie above Min (v, p ), i.e., everybody else in the market is receiving more, and B
feels entitled to pay less than Max (c, p) if (and only if) all external prices lie below Max (c, p ),
i.e., everybody else in the market is paying less. Note that this formulation gives precedence to
an existing contract in the sense that external reference points come into play only when contract
prices are far apart from what’s going on elsewhere. In future work it would be interesting to
explore alternative ways of modeling the interaction between external reference points and prior
contracts.
30
In summary, S feels entitled to receive Max (Min (v, p ), p min) and B to pay Min (Max
(c, p), p max).
Let’s continue with the case where the parties write a date 0 contract [p, p ]. The trading
rule is given by (10) and expected surplus by
(13)
W = ∫ [v − c − θ {Max (Min(v, p), p min) − Min (Max (c, p), p max)}]dF (v, c, p max, p min),
v≥c
v≥p
c≤ p
where we rewrite the distribution function F to reflect the fact that p min and p max can also
depend on the state of the world. Note that (13) coincides with (11) when p min = – ∞, p max =
∞. However, shading costs are higher than before when p < p min or p > p max. Otherwise the
tradeoff between flexibility and rigidity is essentially as in Section III: a large [p, p ] range
makes it more likely that trade will occur, but also leads to more shading.
Matters become more interesting if the parties write no date 0 contract. This is where the
second role of the external reference range [p min, p max] comes in. Previously we supposed
that “no contract” was equivalent to setting p = – ∞, p = ∞, i.e., to a very flexible contract.
However, we now take the view that when the parties bargain at date 1 in the absence of a date 0
contract, they never consider a price below p min or a price above p max because such prices
look unreasonable to outsiders (they might not even be enforced by a court). In other words it is
as if the parties had written an initial contract with p = p min and p = p max: the parties bargain
31
in the intersection of the [c, v] and [p min, p max] ranges, and S feels entitled to receive Min (v,
p max) and B to pay Max (c, p min). Given our simplifying assumption that v ≥ p min and c ≤ p
max whenever v ≥ c, trade will always occur when v > c, and so expected surplus is given by
(14)
W = ∫ [ v − c − θ {Min ( v, p max) − Max (c, p min)}]dF( v, c, p max, p min) .
v≥c
It is easily seen that (14) is different from what is obtained by substituting p = – ∞, p = ∞ in
(13). To underline the point, with external reference points, “no contract” is not the same as a
highly flexible contract.
What determines the choice between writing a date 0 contract and writing “no contract,”
i.e., leaving things to date 1? It is good to write an ex ante contract if p min is small and p max is
large. In particular, if p min = – ∞, p max = ∞, one can do at least as well as “no contract” by
writing a contract with p = – ∞, p = ∞ (compare (13) and (14)); and one can usually do better by
limiting the [p, p ] range. Another case where it is better to write a date 0 contract is if the range
[p min, p max] does not vary with the state of the world – in this case, one can do at least as well
as with no contract by setting p = p min, p = p max.
What about cases where “no contract” is optimal? The leading one is where the range [p
min, p max] is small. Under these conditions trade always occurs when it is efficient (by
assumption, a price can be found in the intersection of the [p min, p max] and [c, v] ranges
whenever v ≥ c), and shading costs are low. For example, if p min = p max = p̂ , “no contract”
achieves the first-best.
32
A concrete example may be useful. Consider Example 1 of Section III. Suppose that p
min = p max = p̂ = 4.5 in s1 and p min = p max = p̂ = 15 in s2. Previously the date 0 contract p
= 9, p = 10 achieved first-best. However, this is no longer true: in s1 B will feel aggrieved that
he pays 9 rather than the (external) price 4.5, and so shades by 4.5 θ; and in s2 S will feel
aggrieved that she receives 10 rather than the (external) price 15, and so shades by 5 θ.
In contrast “no contract” does achieve the first-best. Under no contract, at date 1 B and S
agree on a price of 4.5 in s1 and a price of 15 in s2 – there is nothing to argue about because p
min = p max – and there is no aggrievement or shading in either state.
We conclude by mentioning a possible application of the model to explaining contract
length. Take our example of a wedding that will occur in six months. In this case the range of
reasonable catering prices [p min, p max] is plausibly quite large (there are many different types
of weddings and caterers) and is unlikely to change much over the next six months. According
to our analysis this is a situation where it is better to write an ex ante contract rather than “no
contract.” On the other hand, imagine that the wedding will occur five years from now. There
may be a great deal of uncertainty about future catering costs and future market prices for
catering services (and the two may be highly correlated) and, to the extent that the latter acts as a
reference point for entitlements, fixing a price, or a range of prices, now may create
aggrievement with high probability. It may be better to take a wait-and-see attitude and postpone
contracting. Combining the two cases, i.e., considering a situation of two weddings, one in six
months and the other in five years, yields the beginnings of a theory of contract length.
The above just skims the surface of what is potentially a very interesting and rich set of
issues. We leave the details to future research.
33
VI. RENEGOTIATION
We now relax the no-renegotiation assumption. We will in fact consider three views of
the renegotiation process. We will see that our results are modified, but not entirely changed, by
the possibility of renegotiation. We focus on the models of Sections III and IV.
One view of renegotiation is that, after the contract refinement process illustrated in
Figure I occurs, the parties will always renegotiate to an efficient outcome, and that the parties
will rationally anticipate this. How does this affect our analysis?
Start with the model of Section III. Suppose B and S write a contract consisting of the
price range [p, p ]. Then after the uncertainty about v, c is resolved at date 1 it is possible that v >
c and yet either v < p or c > p . In other words, trade is efficient but won’t occur under the
contract.
According to the first view of renegotiation, B and S will write a new contract. B will
feel entitled to p = c and S to p = v. Total shading is θ (v – c), and net surplus is (1 – θ) (v – c).
Note that renegotiation does not achieve the first-best given θ > 0.
This alters the analysis a little. Take Example 2. Contract (c) is unchanged since no
renegotiation occurs. However, the surplus in contracts (a), (b) rises. Under (a) renegotiation
will take place if s2 occurs. Under (b) renegotiation will take place if s1 occurs. Thus total
surplus under (a), (b) is now
Wa′ = 9 π1 + 10(1 – θ) π2 + 20 π3,
Wb′ = 9(1 – θ) π1 + 10 π2 + 20 π3.
34
Contracts (a) or (b) might now beat contract (c) even if, in the absence of renegotiation, they did
not.
How does this view of renegotiation affect the model of Section IV?
Suppose Δ > δ.
There were two candidates for an optimal contract, (b) and (d). In contract (b) price is fixed and
B chooses the composer. In (d) both price and composer are fixed.
Contract (b) leads to an efficient ex post outcome (apart from shading) and so no
renegotiation will occur. However, contract (d) leads to inefficiency in either s2 or s3, and so
now renegotiation will occur in one of these states. For example, if the contract specifies Bach
and s3 occurs, S will agree to switch to Shostakovich in return for a side payment. The gains
from renegotiation are (Δ – δ), and since there will be aggrievement about how these are split, a
fraction θ of them will be lost to shading.
One problem with this first view of renegotiation is that it makes a strong assumption
about the renegotiation process and entitlements within a state. Consider the switch from Bach
to Shostakovich in s3 in contract (d). It is supposed that S feels entitled at most to 100 per cent
of the surplus from the switch, i.e., to an increase in price from p to p + Δ; and B feels entitled at
most to 100 percent of the surplus from the switch, i.e., to an increase in price from p to p + δ.
But in principle B and S could use the renegotiation to demand even more: S could argue that,
since price is “on the table,” p should move all the way up to v, and B could argue that the price
should move all the way down to c. In other words each party could demand 100 percent of all
the surplus. (Recall the voluntary trade assumption: each party has the option to quit.)
Moreover, they may feel aggrieved if they don’t get this.
We now consider a second view of renegotiation that allows for this possibility. It takes
a “thin end of the wedge” approach. Suppose that if one party proposes a renegotiation
35
involving a price change, then both parties recognize that any price change is now possible.
Assume that the parties will agree to the renegotiation only if both parties are made better off.
Then contract (d) will be renegotiated in s3 only if
(v – c) (1 – θ) ≥ v – c – Δ + δ,
which will not be satisfied if, say, Δ ~
− δ. We see that this second view of renegotiation puts
some friction into the process: it makes it less likely that the parties will switch from an
inefficient composer to an efficient composer.
Note that this second view of renegotiation coincides with the first view for the model of
Section III. The reason is that renegotiation occurs only if the contract leads to no trade, in
which case all the surplus is up for grabs anyway.
In our opinion both of the above views of renegotiation are rosy. Each view supposes
that the possibility of changing price in one state will not affect parties’ feelings of entitlement in
other states. But this is questionable. Return to the model of Section III. Under contract (a), the
price is raised to at least 10 in s2 as a result of renegotiation. Given that price is flexible, why
wouldn’t S feel entitled to a price change in s3? Of course, if S does think this way, then it is as
if the contract specified that the price could be in (at least) the [9,10] range in the first place, and
we are back to contract (c).
In our opinion an intellectually more coherent position is that any flexibility in the trading
price must be built into the initial contract (assuming an initial contract is written at all – see
Section V). That is, one can set p = 9 or p = 10 or p ∈ [9, 10], but one cannot set p = 9 and then
change it to p = 10.
36
Moreover, as we have discussed elsewhere, we believe that this position is consistent
with legal practice and social custom.35 The courts regard contract renegotiations with some
suspicion and may overturn them if they believe that opportunism or duress has played a role.
(Social attitudes and norms often mirror the law.) To this end, the courts require that
renegotiation must be in “good faith,” but, since this is difficult to monitor, they will often
substitute the requirement that the renegotiation can be justified objectively, e.g., the price
increases because the seller is supplying an additional service and her costs have risen.36 In the
model of Section III, no extra service is provided, and so there is no objective justification for a
price change, say from p = 9 to p = 10.37 Similarly, there is no justification for a price change in
the model of Section IV when S switches from Bach to Shostakovich, given that Shostakovich is
not objectively (i.e., verifiably) more costly than Bach.
Under this third view of renegotiation, no renegotiation will occur at all in the models of
Sections III and IV: thus our preceding analysis holds.
VII. SUMMARY AND CONCLUSIONS
In this paper we have developed a theory of contracts based on the view that a contract
provides a reference point for a parties’ trading relationship. The idea is that a contract written
early on when an external measure of the parties’ contributions to the relationship is provided by
competitive markets can continue to govern the parties’ feelings of entitlement later when they
become locked in to each other. The anchoring of entitlements in turn limits disagreement,
aggrievement, and the deadweight losses from shading. We have shown that our theory yields a
trade-off between contractual rigidity and flexibility, provides a basis for long-term contracts in
the absence of noncontractible investments, and throws light on the nature of the employment
37
relationship. We have also shown that an extension of our theory that allows for external
reference points can explain why parties sometimes deliberately write “no contract.”
Our theory is based on strong assumptions. Before considering how these might be
improved on and relaxed, let us say a little more about why we made them. In principle one
could study the trade-off between rigidity and flexibility using more traditional approaches. For
example, standard rent-seeking arguments suggest that a flexible contract that “leaves money on
the table” will generate inefficiency ex post as the parties fight over the surplus.38 Similarly,
influence-cost arguments suggest that mechanisms will be costly given that one party will waste
resources trying to influence the other party’s decision.39 However, these theories suffer from
the following weakness. If, as is usually supposed, ex post trade is perfectly contractible, why
can’t the parties negotiate around the wasteful rent-seeking and influence activities and move
straight to the ex post efficient outcome (assuming symmetric information)?
The main motivation for introducing shading, i.e., for dropping the assumption that ex
post trade is contractible, and for making associated behavioral assumptions, is to avoid this
conclusion.40 In addition our model is tractable and yields intuitive formulae for the costs of
flexibility. For example, in Section IV we saw that letting one party choose the outcome (the
composer) will lead to little inefficiency if the other party is approximately indifferent about the
choice. It is not clear that more traditional approaches generate such a simple (and we think
reasonable) conclusion.
We should make it clear, however, that we are more than happy to consider alternatives
in future work. We mention one. Consider a contract that admits two outcomes, a and b, and
gives the buyer the right to choose between them. If the buyer prefers a and the seller b, the
buyer may have to spend time persuading the seller of the reasonableness of the choice a in order
38
to ensure consummate performance by the seller. These persuasion costs are a plausible
alternative to the shading costs we have focused on. Modeling persuasion costs is not easy but it
is an interesting topic for future research.
Let us turn now to how our behavioral assumptions could be refined and relaxed. Our
model has a number of “black box” features. We have made strong and somewhat ad hoc
assumptions about entitlements, self-serving biases, and shading behavior. We have not derived
these from first principles. Opening up the behavioral black box and showing that these
assumptions are consistent with utility maximizing behavior is an important topic for future
research.
In advance of providing firmer microfoundations, it is worth asking how sensitive our
results are to the particular behavioral assumptions we have made. Take the assumption that
each party feels entitled to the best outcome permitted by the contract. An alternative
specification would be that a party’s entitlement is based on his (rational) expectation of what he
receives in equilibrium [along the lines of Koszegi and Rabin, 2006]. This does not change the
analysis that much as long as entitlements are based on variables specified in the contract, such
as price. Consider Example 2. Under the flexible contract (c), S expects to be paid 9π1 + 10π2 +
9π3, and so is aggrieved in s3 when she receives only 9. Thus there is shading and the first-best
is not achieved.41 Note, however, that the analysis would change significantly if entitlements are
defined relative to net payoffs. Then a contract in which B makes a take-it-or-leave-it offer to S
will yield no aggrievement for S since S’s net payoff is always zero, and so S is never
disappointed. Given that B obtains all the gains from trade, it is plausible (and indeed we have
assumed) that B is not aggrieved either. A model based on these behavioral assumptions would
obviously not generate the trade-off between rigidity and flexibility analyzed here.
39
We have also supposed that shading by B and S is symmetric. As mentioned in the text,
it is easier to think of examples of S’s shading than of B’s shading. One way to generalize our
analysis is to suppose that the buyer and seller’s θ parameters in equations (1) and (2) are
different: if θB < θS, then B both desires and is able to shade less. If 0 < θB < θS, we conjecture
that it will still be optimal to restrict the range of prices in the model of Section III and to fix
price in the model of Section IV. The main change is that seller control of price will be optimal
in the Section III model, and seller control of the composer may be optimal in the Section IV
model even if Δ > δ, i.e., independent contracting is more likely to be optimal when the buyer
can’t shade much.
It should be acknowledged that, if θB = 0, i.e., B does not shade at all, the model as it
stands collapses: the first-best could be achieved by giving S the right to make a take-it-or-leaveit offer to B. S would choose the efficient price-output or price-quality combination, thereby
extracting all the surplus, and, according to standard subgame perfection arguments, B would
accept such an offer. Even in this case, there are ways to rescue the model. One possibility is to
suppose that although B cannot shade, i.e., σB ≡ 0, B experiences aggrievement, i.e., θB > 0 in
equation (1), but B is forced to “eat” the aggrievement rather than being able to shift it to S. The
total deadweight loss from a flexible contract will still be θB aB + θS aS, as in our model. This
approach would bring us closer to that part of the literature that emphasizes loss aversion.42 A
second possibility is to appeal to the ultimatum game literature that argues that B will not accept
S’s offer unless S gives B a reasonable fraction of the surplus. Knowing this, S will allocate
some of the surplus to B, but since S feels entitled to all the surplus she will be aggrieved and
will shade.43
40
A further assumption that we have made is that only the outcome and not the process
matters for people’s feelings of entitlement and well-being. This is strong. Take the
employment contract in Section IV, in which the price is fixed and B has the right to choose the
composer. If B chooses Bach in state s2, which S doesn’t like, S may nonetheless accept and not
feel aggrieved about this choice given that B had the right to make it. In other words, S might
feel differently about the outcome “Bach” if it is the result of a previously agreed process than if
the outcome is arrived at in some other way, e.g., through bargaining.44
Generalizing the model to allow for the role of process is highly desirable. Note that we
do not think that a well designed process will completely eliminate the costs of flexibility. In the
employment example S is likely to accept Bach only if she views B’s choice as “reasonable.”
Convincing S of this may be costly for B; see the above discussion of persuasion costs. Thus our
view is that an appropriately constructed model that allows for the role of process will still
exhibit an interesting trade-off between contractual flexibility and rigidity.
In the Introduction we motivated the paper by pointing out some limitations of existing
models of the firm. We believe that the model developed here can help to overcome some of
those limitations and can be applied to organizational and contract economics more generally.
We have used the model to understand the nature of the employment relationship, and why
parties deliberately write “no contract.” We believe that there are many other possibilities. We
end by mentioning a few of these. First, the model may throw light on the role of the courts in
filling in the gaps of incomplete contracts. The idea is that a contractual term provided by the
parties may affect entitlements whereas one provided by an outsider – the courts – may not. This
may have efficiency consequences. Second, along related lines, the model may help to explain
why parties choose not to index contracts to inflation in order to generate real wage flexibility. If
41
real wages fall because prices rise, this can be blamed on an outsider – e.g., the government –
whereas if nominal wages are reduced by an employer this may generate anger. Finally, the
model of Section IV, extended from two to many people, may help us to understand how
authority should be allocated, i.e., who, out of a group of individuals, should be boss. We
believe that this last application may be a useful step in allowing incomplete contracting ideas to
be applied to the very interesting and important topic of the internal organization of firms.
42
APPENDIX
In this appendix, first, we refine our assumption about what determines a party’s level of
aggrievement; second, we present a result giving circumstances under which we can restrict
attention to contracts where the no-trade price p0 is zero and one party unilaterally chooses the
terms of trade; third, we prove Proposition 2
We assume that when he or she thinks how aggrieved to feel:
A1. A party conjures with mixed strategies in the contractual mechanism (and with
correlated strategies if the mechanism has simultaneous moves).
A2. A party imagines a commitment to trade on the part of the other party, whatever the
outcome of the randomization.
However, no-one believes they can force the other party beyond his or her participation
constraint. Specifically, no-one thinks they can push the other party’s expected payoff below
what he or she could get from simply refusing to trade.
To understand the role played by assumptions A1 and A2, look at a slight variant of the
Example 2 from Section III:
s1
s2
s3
v
9
20
10
c
0
10
9
43
The only change is that in state s3 the (v,c) pair equals (10,9) rather than (20,0). All the analysis
of contracts (a) – (c) from Section III still pertains, except that now they yield expected total
surplus of
Wa = 9 π1 + π3,
Wb = 10 π2 + π3,
Wc = 9 π1 + 10 π2 + (1 – θ) π3,
respectively. As before, in general none of these contracts achieves first-best.
Consider a contract in which p0 ≡ 0 and B chooses p1 from a set of three discrete prices,
{8½, 9½, 10½}. On the face of it, this contract achieves first-best because in each state only one
p1 out of the three allowable prices delivers trade (remember trade is voluntary, so each party has
to be better off than not trading at p0 = 0). But, given assumptions A1 and A2, there will be
aggrievement and hence shading in state s3. When B chooses p1 = 9½, S feels aggrieved that B
didn’t choose a 50:50 lottery between p1 = 9½ and p1 = 10½, with B committing to trade
whatever the outcome of the lottery. The 50:50 odds are such that before the lottery, B would be
no worse off than not trading, given that he values the widget at 10. At a price of 9½, then, S
feels aggrieved by ½. Equally, B feels aggrieved by ½ too! Even though he has the contractual
right to choose p1, and chooses 9½, he would prefer to choose a 50:50 lottery between p1 = 8½
and p1 = 9½, with S committing to trade whatever the outcome: before the lottery, S would be no
worse off than not trading, given that the widget costs her 9. In aggregate, shading in state s3
amounts to (½ + ½)θ, which is the same as under contract (c) in Section III. Worse, in state s1,
44
when B chooses p1 = 8½, S feels aggrieved that B didn’t choose a 50:50 lottery between p1 = 8½
and p1 = 9½, with B committing to trade whatever the outcome, so that in this state there is
shading. Similarly, there is shading in state s2. All in all, our contract is strictly dominated by
contract (c).
If the no-trade price p0 varies with the strategies, and cannot simply be normalized to
zero, then it is a little less obvious how participation constraints should be factored into the
parties’ thinking. Consider a contract in which B chooses (p1, p0) from a set of three pairs: (9, 1),
(9½, ½) or (10, 0). What would happen if either party refused to trade? B would minimize p0 by
choosing (p1, p0) = (10, 0). So, in reckoning how low they can push the other’s payoff, each
party thinks in terms of a default no-trade price of p0* = 0. Precisely, p0* is the right-hand side
(RHS) of S’s participation constraint; and (– p0*) is the RHS of B’s participation constraint.
Knowing this, we can invoke assumptions A1 and A2 to calculate aggrievement levels in each
state. In particular, in state s3, B chooses (p1, p0) = (9½, ½) but is actually aggrieved by ½
because he would prefer to choose (p1, p0) = (9, 1), with S committing to trade – this would not
violate S’s participation constraint given p0* = 0. S is also aggrieved by ½ because she would
prefer B to choose (p1, p0) = (10, 0) and commit to trade. In aggregate, shading in state s3
amounts to (½ + ½)θ – the same as under contract (c). One can show that our contract delivers
the same total surplus as contract (c) in states s1 and s2 too.45
But this begs the question: what might a more general contract be able to achieve?
Below, we present Proposition 4 giving circumstances under which non-zero values of p0 do not
help and we can restrict attention to contracts where one party unilaterally chooses the terms of
trade.
45
A general contract C can be viewed as a stochastic mechanism mapping from a pair of
messages β and σ, reported by B and S respectively, onto either a pair of (trade, no-trade) prices,
(p1, p0), or onto simply a no-trade price, p0. In other words, following the report of messages β
and σ, there is an exogenous lottery to determine (i) if trade is allowed or not; (ii) the terms of
trade/no-trade. If trade is allowed then it occurs if and only if both parties want it (at price p1).
Otherwise, there is no trade (at price p0). (Remember we are assuming no renegotiation, so if the
mechanism specifies no trade then that outcome is final.)
In effect, then, a mechanism allows for probabilistic trade – a surrogate for fractional
trade (our widget is assumed to be indivisible).
Under contract C, p* (the default no-trade price used to determine the RHS’s of the
parties’ participation constraints) is the value of the zero-sum game over the p0’s specified in the
mechanism – under the supposition that one or other party “quits”, i.e. always chooses to veto
~
trade, so that the specified p1’s are irrelevant. Let ( β * , σ~ * ) be the (possibly mixed) equilibrium
strategies of this zero-sum message game.
We consider, for each state (v, c), a (possibly mixed strategy) subgame perfect
equilibrium of the game induced by contract C. Let q(v, c) be the probability of trade in
equilibrium, and p(v, c) the expected payment from B to S. Since trade is voluntary, q(v, c) > 0
only if v ≥ c. If there is more than one equilibrium, we pick the one that maximizes q(v, c).
No party is worse off than if he or she quit:
LEMMA
vq(v, c) – p(v, c) ≥ – p0*,
p(v, c) – cq(v, c) ≥ p0*.
46
Proof : Consider B. In state (v, c) he could deviate from his equilibrium message~
reporting-cum-trading strategy to report β * and always refuse to trade. Were he to do so,
his payoff would (weakly) drop from vq(v, c) – p(v, c) to, say, ( − p̂0 ), where p̂0 is the
ensuing (expected) no-trade price. But p̂0 cannot be strictly more than p0*, because
~
σ~* is a best reply to β * for S in the zero-sum game over the p0’s. This proves the first
inequality in the Lemma. The second follows symmetrically – reversing the roles of B
and S.
QED
For future reference, let H ⊆ [0,1] × R denote the convex hull of (0, 0) and all pairs {q(v,
c) , p(v, c) – p0*}, where, notice, we are netting prices by subtracting p0*. In the space of
quantity (q) and net price (p – p0*), the set H might look as follows:
[FIGURE III HERE.]
Invoking assumptions A1 and A2, in state (v, c) we define B’s [resp. S’s] “aspiration
level” b(v, c) [resp. s(v, c)] to be his [resp. her] maximum payoff across all correlated message
pairs and trading rules subject to the constraint that S [resp. B] gets no less than p0* [resp. (–
p0*)].
In particular, B and S each imagine that they could jointly precommit to the (mixed)
message-cum-trading equilibrium strategies pertaining to some other state, or some convex
combination thereof. Hence
47
(i)
b(v, c) ≥ max {vq – p| (q, p − p0 *) ∈ H and p − cq ≥ p0 * },
q, p
(ii)
s (v, c) ≥ max {p – cq| (q, p − p 0 *) ∈ H and vq − p ≥ − p 0 * }.
q, p
Note that, thanks to the Lemma,
b(v, c) ≥ vq(v, c) − p (v, c) ,
s (v, c) ≥ p(v, c) − cq(v, c) ;
i.e., aspiration levels are at least as high as equilibrium payoffs.
In each state (v, c), once equilibrium play is over, B shades by reducing S’s payoff down
to
p (v, c) − cq(v, c) − θ {b(v, c) − [vq(v, c) − p (v, c)]}.
And S shades by reducing B’s payoff down to
vq (v, c) − p (v, c) − θ {s (v, c) − [ p (v, c) − cq (v, c)]} .
Hence, in the special case θ = 1 (the case considered in Proposition 4 below), total surplus in this
equilibrium equals
48
(iii)
2( v − c ) q ( v , c ) − b ( v , c ) − s ( v , c ) .
Now consider contract Ĉ, in which p0 ≡ 0 and one party (say B – it doesn’t matter who)
chooses from a set of exogenous lotteries, each corresponding to a different point (q, p) ∈ H :
with probability q, trade is allowed at p1 =
p
,
q
with probability 1 – q, trade is not allowed.
At first sight contract Ĉ may look a little strange, but that is because it is dealing with
probabilistic trade. Note that for q = 1 – corresponding to the right-hand edge of the set H in
Figure III – contract Ĉ is nothing more than our “standard” contract in which B chooses a trading
price p1 from an interval [p, p ]. To put this another way, if the upper and lower edges of the set
H in Figure III were linear rather than piecewise linear, H would correspond to a standard
contract.
PROPOSITION 4.
Suppose θ = 1. Then contract Ĉ yields at least as much total surplus in
each state as does contract C.
Proof. Under contact Ĉ, in state (v, c), B’s aspiration level is
bˆ(v, c) = max {vq – p| (q, p) ∈ H and p − cq ≥ 0 }
q, p
(iv)
≤ p0* + b(v, c)
49
by (i). And S’s aspiration level is
sˆ(v, c) = max {p – cq| (q, p) ∈ H and vq − p ≥ 0 }
q, p
(v)
≤ – p0* + s(v, c)
by (ii). In each state (v, c), B chooses the lottery corresponding to a point
(q, p) ∈ H to maximize his net payoff (i.e. net of S’s shading), taking into account that S
may not be willing to trade at p1 =
p
for q ≠ 0. That is, given θ = 1, B chooses
q
( q, p ) ∈ H to maximize
vq − p − {sˆ(v, c) − [ p − cq]} subject to p – cq ≥ 0.
If v > c, in effect B will maximize the probability of trade, q, subject to ( q, p ) ∈ H for
some p ≥ cq. Call this maximum qˆ (v, c) . But from the definition of the set H,
( q (v, c ), p (v, c ) − p 0 *) ∈ H ;
and from the second inequality in the Lemma,
p(v, c) − p0 * ≥ cq(v, c) .
50
Hence qˆ (v, c) is at least q(v, c) whenever v > c.
If v < c, B will choose (q, p) = (0, 0); in this case set qˆ (v, c) = 0.
Combining these two cases, we have
(vi)
(v − c)[qˆ (v, c) − q(v, c)] ≥ 0 in all states (v, c).
Just as total surplus in state (v, c) was given by expression (iii) under contract C, so too
under contract Ĉ it is given by
(vii)
2(v − c)qˆ (v, c) − bˆ(v, c) − sˆ(v, c).
But, making use of inequalities (iv) – (vi), we see that the expression in (vii) is no less
than that in (iii).
QED
In words, Proposition 4 states that, without loss of generality, p0 can be normalized to
zero and one party (B, say) can be given control over the terms of trade. The subset H of
[0,1] × R is the “design variable”. It is this set that the contract specifies, taking any shape (along
the lines of that in Figure III) – but it must be convex and, for q = 0, come to a point at the origin
(i.e., when q = 0, p = p0* = 0).
Of course, the weakness of this result is that it applies only to the limit case θ = 1.
However Proposition 4 is suggestive of other more general results. For example, it may be quite
51
general that assumptions A1 and A2 are enough to allow us to ignore the possibility of non-zero
values of p0. The Proposition as it stands may apply if θ is close enough to 1. And for lower
values of θ, somewhat more complex allocations of control rights over the terms of trade (not
merely giving unilateral control to either B or S) may turn out to be optimal. All this awaits
further research. Meantime, the Proposition gives us some reassurance that, by restricting
attention (as we do in Section III) to standard contracts in which p0 ≡ 0 and B chooses p1 from an
interval [p, p ], we are not missing out on some sophisticated contract that achieves first-best.
We now turn to Section IV, and the proof of Proposition 2. Think of composers as lying
in the [0, 1] interval, with λ = 0 corresponding to Bach and λ = 1 corresponding to Shostakovich.
In state s2, the value of composer λ to B is v – λΔ, and the cost to S is c – λδ. (That is, λ is
equivalent to a convex combination of Bach and Shostakovich.) In state s3, the value of λ to B is
v – (1 – λ) Δ, and the cost to S is c – (1 – λ) δ. We restrict attention to deterministic contracts in
which the no-trade price is zero.
As a preliminary, we should observe that if no music is played in state s4 then in the
other three states the first-best could be achieved using a contract that fixes the price at v and has
B choose the composer. In state s2, B would choose λ = 0 (as first-best requires) but there would
be no aggrievement on the part of S since at price v no other composer would satisfy B’s
participation constraint. Likewise in state s3, B would choose λ = 1 and there would be no
aggrievement. In state s1, there would be no aggrievement either (B could choose any
composer). At this high price B would be unwilling to trade in state s4. Overall, expected total
surplus would be (1 – π4)(v – c). For small enough π4, this would be the optimal contract. But
we see this as a peculiar case, which we can later confirm is ruled out if π4/(1 – π1) is above the
52
lower bound in Proposition 2. (Incidentally, this is the role of state s4 in the model when Δ > δ.
State s1 plays an analogous role when Δ < δ.)
From now on, we suppose that music is played in state s4, at price p4, which must lie at or
below B’s value, v – Δ. It is straightforward to confirm that in states s1, s2 and s3 music is also
played under an optimal contract. In state s2, suppose composer λ2 is played at price p2. We
restrict attention to symmetric contracts, so that in state s3 composer 1 – λ2 is played at price p2.
The method we will use to characterize an optimal contract is to include in our
mathematical program only those inequality constraints that are critical. At the end, we will
need to confirm that the (many) missing constraints are satisfied. In particular, for now we shall
ignore the question of who controls the choice of composer and price.
Let a2 be the total level of aggrievement (B’s plus S’s) in state s2 when λ2 is played at
price p2. (By symmetry, a2 is also the total level of aggrievement in state s3.) Now S may prefer
to switch from composer λ2 to composer 1 – λ2 at the same price p2 (which is admissible in the
contract, since that is what occurs in state s3), to reduce her costs from c – λ2δ to c – (1 – λ2)δ –
unless this switch would violate B’s participation constraint, v – (1 – λ2)Δ – p2 ≥ 0, in which case
the best S could wish for is to switch to composer
v − p2
at price p2 and reduce her costs from c
Δ
⎛ v − p2 ⎞
– λ2δ to c − ⎜
⎟δ (the intermediate composer is equivalent to a suitable lottery of
⎝ Δ ⎠
composers λ2 and 1 – λ2, and we appeal to assumptions A1 and A2) . Thus
(viii) a2 ≥ δ min {1 – 2λ2,
v − p2
– λ2}.
Δ
53
Let a1 be the total level of aggrievement in state s1, where all composers cost c and yield
value v. We know p2 cannot strictly exceed v; but suppose p2 ≥ v – Δ. Then, given that p4 ≤ v –
Δ, any price in the interval [v – Δ, p2] is contractually feasible (with a suitable lottery). Invoking
assumptions A1 and A2, we therefore have
(ix)
a1 ≥ p2 – v + Δ.
If p2 < v – Δ the inequality (ix) still holds: the RHS is negative.
If a4 is the total level of aggrievement in state s4, consider the relaxed program: Choose
λ2, p2, a1, a2 and a4 to maximize
(x)
W ≡ π1 [v – c – θa1]
+ (1 – π1 – π4)[v – c – λ2(Δ – δ) – θa2]
+ π4 [v – c – Δ + δ – θa4]
subject to (viii), (ix) and the constraint that total aggrievement is always nonnegative:
(xi)
a1 ≥ 0 , a2 ≥ 0 , a4 ≥ 0.
In a solution to this relaxed program, the tighter of the lower bound constraints on a1 will
bind. Likewise for a2. And a4 = 0.
Now consider the role of p2, which affects W in (x) only via a1 and a2. Via a1, the slope
of W w.r.t. p2 is (–π1θ) if p2 ≥ v – Δ, and is zero otherwise. Via a2, the slope of W w.r.t. p2 is at
54
most (1 – π1 – π4)θδ/Δ, and is zero if p2 < v – (1 – λ2)Δ. Hence, from the lower bound on π1/(1–
π4) in Proposition 2, p2 should be as small as possible. However, there is no value in pushing p2
below v – Δ since this would not affect W.
Next, consider the role of λ2 ∈ [0, 1], given p2 = v – Δ. λ2 affects W only through the
middle term in (x), and via a2. Values of λ2 above ½ are clearly not optimal. If Δ > (1 + 2θ)δ, λ2
should be zero; whereas if Δ < (1 + 2θ)δ, λ2 should equal ½. These are two conditions that
appear in Proposition 2.
Where does this leave us? On the one hand, if Δ > (1 + 2θ)δ we can implement the above
solution to the relaxed program (viz., p2 = v – Δ, λ2 = 0, a1 = a4 = 0 and a2 = δ) using a contract
in which the price is fixed at v – Δ and B chooses the composer. (Actually, any fixed price
between c and v – Δ would yield the same W.) On the other hand, if Δ < (1 + 2θ)δ we can
implement the solution (viz. p2 = v – Δ, λ2 = ½ and a1 = a2 = a4 = 0) using a contract in which the
price is again fixed at v – Δ, but so too is the composer, at λ2 = ½. (Actually in this latter case, it
doesn’t matter which composer is fixed; it could instead be Bach, λ2 = 0.) The fact that in all
cases the solution to the relaxed program can be implemented using some contract vindicates our
earlier decision to omit the other inequality constraints.
Proposition 2 is proved.
QED
The reader may wonder why the auxiliary condition π1/(1 – π4) ≥ δ/(Δ + δ) is needed in
Proposition 2. Consider the following variant to a simple employment contract: the basic wage
is v – Δ in return for performing λ = ½, but B can pay a supplement Δ/2 in return for asking S to
perform some other λ in [0,1]. In states s1 and s4, B will pay the basic wage only and λ = ½ will
be performed. And in state s2 [resp. s3], B will pay the supplement and ask S to perform λ = 0
55
[resp. λ = 1] – note that without factoring in the impact of shading by S, B is indifferent about
paying the supplement in states s2 and s3, but it can be checked that S shades even more if B
does not pay it. In each of states s1, s2 and s3, S would wish that B had paid the supplement and
asked her to perform λ = ½. So she is aggrieved by Δ/2 in state s1, and by δ/2 in states s2 and s3.
(In state s2, it would be unreasonable for S to wish that B had asked her to perform any λ > ½
because that would violate B’s participation constraint. Similarly, in state s3, any λ < ½ would
be unreasonable.) In state s4, S cannot wish that B pay more than his value, v – Δ, so there is no
aggrievement. Overall, then, the contract yields expected total surplus
π1 [v – c – θ(Δ/2)] + (1 – π1 – π4)[v – c – θ(δ/2)] + π4 [v – c – Δ + δ].
This can be strictly greater than the expected total surplus yielded by any simple employment
contract if the auxiliary condition π1/(1 – π4) ≥ δ/(Δ + δ) is not satisfied. For example, if v = 20,
c = 10, Δ = 6, δ = 2, π1 = 0.1 and π4 = 0.5, then the above contract yields an expected total surplus
8 – (7θ/10), whereas the best simple employment contract (viz., B chooses any λ, and the wage is
fixed at v – Δ) yields only 8 – (4θ/5).
HARVARD UNIVERSITY
EDINBURGH UNIVERSITY AND LONDON SCHOOL OF ECONOMICS
56
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61
ENDNOTES
1. For up-to-date syntheses of the classical view, see Bolton and Dewatripont [2005] and Shavell [2004].
2. See, e.g., Grossman and Hart [1986] and Hart and Moore [1990].
3. For a discussion, see Holmstrom [1999].
4. See, e.g., Maskin and Tirole [1999] and the response in Hart and Moore [1999].
5. One obvious possibility is to introduce asymmetric information. To date such an approach has not been very fruitful in the theory
of the firm. But see Matouschek [2004].
6. We do not go as far as some of the recent incomplete contracting literature that supposes that ex post trade is not contractible at all
[see, e.g., Baker et al., 2006]. One problem with supposing that ex post trade is noncontractible is that it is unclear how one party gets an action
carried out except by doing it himself. As will be seen, our approach does not suffer from this difficulty.
7. The perfunctory and consummate language is taken from Williamson [1975, p. 69].
8. For a discussion and examples, see Goldberg and Erickson [1987, p. 388].
9. Since a court can determine whether trade took place, any payments that B has promised S conditional on trade must be made: if
not, S would sue for breach of contract. In other words, payments are part of perfunctory performance. (The speed with which payments are
made, however, may be part of consummate performance.)
10. This idea is consistent with the large behavioral economics literature that has examined altruism, reciprocity, and retaliation. For
example, in the ultimatum game [see, e.g., Guth et al., 1982], a suggested split of surplus by the proposer that is seen as “greedy” will often elicit
retaliation in the form of rejection by the responder, even though this is costly for the responder. See Camerer and Thaler [1995] for a discussion,
and Andreoni et al. [2003] for experimental evidence for the case where the responder can scale back the level of trade rather than rejecting trade
entirely. Other important works on reciprocity and retaliation include Akerlof [1982], Akerlof and Yellen [1990], Fehr and Schmidt [1999],
Rabin [1993], Fehr et al. [1997], and Bewley [1999]; for surveys see Fehr and Gachter [2000] and Sobel [2005]. MacLeod [2003] models the
role of retaliation in sustaining accurate assessments of worker performance. Direct empirical evidence of retaliation by employees or contractors
in response to “bad treatment,” in the form of poor performance, negligence, or sabotage, can be found in Lord and Hohenfeld [1979], Giacalone
and Greenberg [1997], Greenberg [1990], Krueger and Mas [2004], Mas [2006a], and Mas [2006b].
11. Note that the experimental evidence of Falk et al. [2003] is consistent with the idea that whether a person feels well treated
depends not only on the outcome that occurs but also on what other outcomes were available [see also Camerer and Thaler, 1995].
12. The notion of a reference point has played an important role in the recent behavioral economics literature, including that concerned
with contractual relationships. Kahneman et al. [1986] provide evidence that for transactions between firms and consumers customers use past
prices as a reference point for judging the fairness of a transaction. See also Okun [1981], Falk et al. [2006], Frey [1997, Chapter 2], and Gneezy
and Rustichini [2000] for related ideas. Akerlof and Yellen [1990] consider the importance of reference groups in the determination of a fair
62
wage. Benjamin [2006] analyzes the implications of reference points for optimal incentive schemes, and Carmichael and MacLeod [2003] for the
hold-up problem. Our paper owes a lot to the above literature, but differs from it in supposing that a contract governing a transaction is a
reference point for the transaction itself.
13. For example, each party may feel that he brings special skills to the trading relationship, or that he has taken an (unmodeled)
action that has contributed to this relationship, and that he deserves to be rewarded for these things. Virtue is in the eye of the beholder and so
each party may exaggerate the importance of his own contribution. For contributions to the self-serving bias literature, see, e.g., Hastorf and
Cantril [1954], Messick and Sentis [1979], Ross and Sicoly [1979], and Babcock et al. [1995]; and for discussions, see Babcock and Loewenstein
[1997] and Bazerman [1998, pp. 94-101]. For models of self-serving biases, see Rabin [1995] and Benabou and Tirole [2006]. Note that
conflicting notions of entitlement may also arise because of differences in information about the total surplus available. See Ellingsen and
Johannesson [2005].
14. Embodied in the above formulation is the idea that aggrievement is measured relative to gross payoffs,
not net payoffs. Arguably, parties might measure aggrievement in terms of net payoffs, in the sense that
aggrievement equals the difference between the maximum payoff a party could have achieved, taken over all
contractually feasible outcomes, and his net payoff. If θ < 1, this alternative formulation leads to a fixed point in net
payoffs. The analysis does not change substantively.
15. This assumption is not crucial; see footnote 27.
16. For further discussion and examples of seller shading, see Goldberg and Erickson [1987, p. 388], Lord and Hohenfeld [1979],
Giacalone and Greenberg [1997], Greenberg [1990], Krueger and Mas [2004], and Mas [2006b]; and for a vivid account of how getting less than
what you think you are entitled to can lead to feelings of aggrievement and the desire to shade, see Kidder [1999]. One approach to solving the
problem of seller shading is for the buyer to withhold some part of his payment until after the transaction is complete, i.e., to offer a bonus or tip
to the seller for consummate performance. One well known problem with this is that in a one-shot situation the buyer has an incentive to claim
that performance is not consummate even if it is in order to avoid paying the tip. Note that this is true even if the seller has an opportunity to
retaliate after the nonpayment of the tip, as long as the shading or retaliation parameter θ < 1 (the buyer loses θt but gains t, where t is the tip). In
practice tips are paid, probably because of social norms (or because the relationship is repeated). However, it is unclear that a norm-induced tip
will elicit consummate performance: if the seller knows that she is going to get somewhere between 15 percent and 20 percent pretty much
regardless of performance, her incentive to provide consummate performance is limited. For a more positive view of the effectiveness of tips and
bonus schemes, see Scott [2003].
17. If B’s value of 100 and S’s cost of zero were objective (i.e., verifiable), the parties might well agree that the fair outcome is to
split the difference and set p = 50. We have in mind a more complex situation, where, because value and cost are observable but not verifiable,
there is some flexibility in how the parties interpret these variables; this opens the door to conflict. While we do not formalize this notion of
flexibility or fuzziness, it would clearly be desirable to do so in future work.
63
18. Note the role of the assumption that θ ≤ 1. If θ > 1, the shading costs are so large that the parties will not trade at all at date 1 in
the absence of a date 0 contract. Although we rule out the case θ > 1, this does not mean that it is uninteresting.
19. Note that we are ignoring “efficiency wage” considerations in our analysis. Regardless of date 0 market conditions, B might well
feel that it makes sense to offer S a price in excess of cost in order to encourage better performance [see, e.g., Shapiro and Stiglitz, 1984].
However, note that efficiency wage ideas are not inconsistent with our approach. Our view is that, whatever the level of the price, it still makes
sense for B and S to fix price in advance in order to avoid argument about the right price later.
20. However, legal enforcement may be important in ensuring perfunctory performance, including that B pays S for goods or services
received at date 1.
21. This assumption is taken from Hart and Moore [1988].
22. On agreements to agree generally, and on whether they are binding in particular, see, e.g., Corbin [1993, Chapters 2 and 4] and
Farnsworth [1999, pp. 207-222].
23. Of course, under the agreement-to-agree interpretation, p0 is necessarily zero.
24. See Maskin [1999].
25. It is worth noting that, if third parties are permitted, the first-best can be achieved in Example 2. Consider a contract that fixes the
trade price and makes both B and S pay a large amount to a third party in the absence of trade (i.e., the no-trade price is large and positive for B
and large and negative for S). This leads to trade in all states, and no aggrievement, since the consequences of not trading are dire. However, this
arrangement works only because trade is efficient in every state. In a more general example where trade is efficient in some states but not others,
third parties do not guarantee the first-best. In what follows we ignore third parties.
26. In reality there is a continuum of possibilities between completely verifiable states at one extreme and completely subjective
states at the other. The more objective the state the easier it is to avoid aggrievement by writing a state-contingent contract. We have simplified
matters drastically by considering only the two extremes: verifiability, which permits the writing of state contingent contracts, and
nonverifiability, which does not. In future work it would be desirable to explore the territory in between.
27. In the model of this section we have supposed that shading occurs only in the presence of trade. The no-trade/no-shade
assumption can easily be relaxed, however. Proposition 1 continues to hold, but now it is easier to find cases where the first-best cannot be
achieved. (In the following example we drop the assumption that a party feels entitled to no more than one hundred percent of the gains from
trade.) Suppose that there are two states: in s1, v = 40, c = 30 (trade is efficient); in s2, v = 10, c = 20 (trade is inefficient). Under our previous
assumptions a contract with p = 35, say, achieved the first-best since there was nothing to argue about in s1 and there was no trading or shading in
s2. Now any price 30 ≤ p ≤ 40 that allows trade in s1 causes S to be aggrieved in s2 (S loses money because B refuses to trade) and leads to
shading. Thus the first-best cannot be achieved.
28. See, e.g., Corbin [1993, Chapter 4.3], Ben-Shahar [2004, pp. 424-425].
64
29. It is worth revisiting our assumptions about aggrievement and shading at this point. Take contract (b), where price is fixed and B
chooses the composer. Why is S aggrieved in s2 or s3 when B chooses the high cost composer? In Sections 2 and 3 we offered one justification:
S may feel that she has contributed to surplus and should be rewarded with the low cost composer. Here we offer an alternative justification
based on subjective views of valuation. Suppose that the state is s2. S can tell herself the following. It’s true that B thinks that Bach is worth Δ
more to him than Shostakovich, while the incremental cost to me is only δ, and B therefore feels justified in choosing Bach. However, I, S, think
that, even though B doesn’t realize this, the true value of Bach is close to v – Δ. (Self-serving biases are behind this belief.) Given this, I do not
think that B’s request is reasonable, and if he insists on Bach, I will respond by shading on performance.
30. But see Wernerfelt [1997].
31. This notion of independent contracting differs from Simon’s. Simon views independent contracting as corresponding to the case
where the price and composer are both fixed.
32. The distinction between an employee and an independent contractor shouldn’t be taken too literally. An independent contractor
will sometimes do what a buyer tells her to, and an employee sometimes will not. Still, in general terms the notion that an employee is subject to
the authority of a boss whereas an independent contractor is not seems valid. For a discussion, see, e.g., Coase [1937, p. 403-404].
33. For an interesting discussion of this kind of phenomenon and its implications for labor market practices, see Akerlof and Yellen
[1990]. For evidence on the importance of external reference points and their effects on “shading” behavior by employees, see Babcock et al.
[1996].
34. We do not suppose that the [p min, p max] range is verifiable, however; hence it cannot be made part of an enforceable contract.
35. See Hart and Moore [2004].
36. See Restatement (Second) of Contracts, Section 84(a)[1979]; Farnsworth [1999, pp. 276-295]; Jolls [1997, pp. 228-301]; Muris
[1981, particularly p. 530]; and Shavell [2005].
37. In fact, without some constraint on price changes, a long-term contract would have little meaning. Almost every contract is
incomplete in the sense that some ex ante noncontracted-for cooperation is required ex post for the contract to succeed. If each party can demand
a large sidepayment for that cooperation that is completely unrelated to costs – you want a glass of water that will cost you $1,000 – the initial
contract will be vitiated.
38. See, e.g., Tullock [1967].
39. See, e.g., Milgrom [1988].
40. But see Bajari and Tadelis [2001]. Bajari and Tadelis consider a model of the trade-off between flexibility and rigidity, where
rigidity, in the form of a fixed price, is good because it encourages efficient cost reduction by the seller S, but bad because it impedes ex post
adjustment. Our model has somewhat similar ex post characteristics to Bajari and Tadelis’s, but ignores ex ante incentives for cost minimization.
41. A subtlety that arises under the Koszegi-Rabin formulation is that in some states parties will receive more than their entitlements.
The model goes through as long as parties who receive more than their entitlements do not “over-provide” consummate performance to the point
65
where this makes up for their perfunctory performance when they receive less than their entitlements. For some experimental/empirical evidence
consistent with such an asymmetry, see Charness and Rabin [2002], Offerman [2002], and Mas [2006a].
42. See, e.g., Kahneman and Tversky [1979].
43. There are at least two other ways to avoid the “first-best” conclusion when B can’t shade. One is to suppose that B must make an
ex ante noncontractible investment, so that complete hold-up by S is undesirable. A second is to suppose that S is wealth constrained and so
cannot compensate B in advance for the 100 percent of the surplus that S will obtain ex post.
44. We are grateful to Birger Wernerfelt and Klaus Schmidt for discussions on this issue.
45. If we do not invoke assumptions A1 or A2, our contract looks attractive. It could be argued that since
B is choosing the pair (p1, p0), he is never aggrieved. And the contract has been designed so that in state s3 S is
aggrieved by only ½. She would increase her payoff by ½ if, instead of B choosing (p1, p0) = (9½, ½) leading to
trade at p1 = 9½, B were to choose (p1, p0) = (9, 1) leading to no trade at p0 = 1; equally, S would increase her payoff
by ½ if B were to choose (p1, p0) = (10, 0) leading to trade at p1 = 10. Shading in state s3, then, amounts to ½θ – half
that under contract (c). Likewise, in state s1 there is shading of ½θ because S would prefer B to choose (p1, p0) =
(9½, ½) leading to trade at p1 = 9½; and in state s2 there is shading of θ because S would prefer B to choose (p1, p0)
= (9, 1) leading to no trade at p0 = 1. Overall, expected total surplus from our contract is (9 – ½θ) π1 + (10 – θ) π2 +
(1 – ½θ) π3 – which means that if, for example, π1 + 2π2 < π3 and θ = 1, our contract would dominate contracts (a),
(b) and (c) in Section III.
66
0
1
│_______________________________________________│
Parties meet and
Contract refined after
contract.
resolution of uncertainty.
Trade occurs and degree of
consummate performance
determined.
FIGURE I
67
Bach
Shostakovich
s1 (Prob π1)
(v,c)
(v,c)
s2 (Prob (1- π1- π4)/2) s3 (Prob (1- π1- π4)/2)
(v,c)
(v-Δ,c-δ)
(v-Δ,c-δ)
(v,c)
FIGURE II
68
s4 (Prob π4)
(v-Δ,c-δ)
(v-Δ,c-δ)
p - p0*
H
1
0
FIGURE III
69
q